Lie Transform Based Polynomial Neural Networks for Dynamical Systems Simulation and Identification

In the article, we discuss the architecture of the polynomial neural network that corresponds to the matrix representation of Lie transform. The matrix form of Lie transform is an approximation of general solution for the nonlinear system of ordinary differential equations. Thus, it can be used for simulation and modeling task. On the other hand, one can identify dynamical system from time series data simply by optimization of the coefficient matrices of the Lie transform. Representation of the approach by polynomial neural networks integrates the strength of both neural networks and traditional model-based methods for dynamical systems investigation. We provide a theoretical explanation of learning dynamical systems from time series for the proposed method, as well as demonstrate it in several applications. Namely, we show results of modeling and identification for both well-known systems like Lotka-Volterra equation and more complicated examples from retail, biochemistry, and accelerator physics.